Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors.
In general, let be a first-order differential operator acting on a vector bundle over a Riemannian manifold .
If
with being the Laplacian of , is called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part of must equal the Laplacian.
Examples
1. is a Dirac operator on the tangent bundle over a line.
2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin 1⁄2 confined to a plane, which is also the base manifold. Physicists generally think of wavefunctions ψ: R2 → C2 which they write
where x and y are the usual coordinate functions on R2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written
where σi are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra.
Solutions to the Dirac equation for spinor fields are often called harmonic spinors[1].
3: The most famous Dirac operator describes the propagation of a free electron in three dimensions and is elegantly written
using Einstein's summation convention and even more elegantly as
using the Feynman slash notation.
4: There is also the Dirac operator arising in Clifford analysis. In euclidean n-space this is
where
is an orthonormal basis for euclidean n-space.